Almkvist's unimodality conjecture for two-variable Hilbert functions

Let $m,n$ be positive integers, let $R={\mathbb Q}[x_1,\ldots,x_n]$ with its standard grading, and write $e_i=e_i(x_1,\ldots,x_n)$ for the elementary symmetric functions and $e_i(m)=e_i(x_1^m,\ldots,x_n^m)$. Define

$$A(m,n)=\frac{{\mathbb Q}[e_1,\ldots,e_n]}{(e_1(m),\ldots,e_n(m))}$$

and let $H(m,n)=(H(m,n)_k)_{k=0}^d$ be the coefficient sequence of its Hilbert polynomial.

Almkvist's conjecture. For each $m$, the Hilbert function $H(m,n)$ is unimodal for all $n\geq 11$. Moreover, if $m$ is even, then $H(m,n)$ is unimodal for all $n$.

These Hilbert functions also enumerate a two-parameter family of integer partitions and are related to the unimodality of graded Artinian complete intersections. The conjecture is attributed to G. Almkvist and is presented here without a stated resolution.